Finding ⟨Lx⟩ from ⟨L²⟩ and ⟨Lz⟩, and changing basis between operators

[broegger]

Two quick ones :)

Hi, two questions:

1) How can I find the expectation value of the x-component of the angular momentum, $$\langle L_x \rangle$$, when I know $$\langle L^2 \rangle$$ and $$\langle L_z \rangle$$?

2) Say, I have a state $$|\Psi \rangle$$ and two operators A and B represented as matrices. Now, $$|\Psi \rangle$$ is given as a linear combination of the eigenstates of A and I want to express them as a linear combination of the eigenstates of the operator B instead. How do I do that?

Thanks :)

 

[vanesch]

1) How can I find the expectation value of the x-component of the angular momentum, $$\langle L_x \rangle$$, when I know $$\langle L^2 \rangle$$ and $$\langle L_z \rangle$$?

I don't think you can ! I think you can if you are in an EIGENSTATE of L^2 and Lz, but not if you only know the expectation values, because I think I can make up two different wave functions with same expectation for L^2 and Lz and different expectation for Lx...

cheers,
Patrick.

 

[broegger]

Oh, they ARE eigenstates of both L^2 and Lz. Sorry. What do I do? :)

 

[vanesch]

Oh, they ARE eigenstates of both L^2 and Lz. Sorry. What do I do? :)

I think you will always find 0, no ? Because of the cylindrical symmetry of these states ?

The reason is that you can write Jx as 1/2 (J_+ + J_-) and these, acting on a state |j,m> will give you |j,m+1> and |j,m-1>, so
<j,m| Jx |j,m> = 0, no ?

cheers,
patrick.

 

[broegger]

I don't quite get it. What does the |J>'s represent?

 

[vanesch]

I don't quite get it. What does the |J>'s represent?

Sorry, I should have written L. J stands in general for an angular momentum, L for an orbital angular momentum and S for a spin angular momentum. One usually uses the notation J if it doesn't matter (as is the case here) whether it is orbital or spin angular momentum one talks about.

I would think you are aware of J+ and J- (or L+ and L-), the ladder operators of angular momentum ?

cheers,
Patrick.

 

[vanesch]

2) Say, I have a state $$|\Psi \rangle$$ and two operators A and B represented as matrices. Now, $$|\Psi \rangle$$ is given as a linear combination of the eigenstates of A and I want to express them as a linear combination of the eigenstates of the operator B instead. How do I do that?

I guess you mean that you work in a certain basis, in which you get to know the components of A (matrix) and B (matrix).
However, there is an ambiguity in the way you know psi: if you know its projections on the eigenvectors of A, you don't necessarily know its components in the basis in which A is given, because eigenvectors are only defined upto a complex multiplication factor ! So depending on how these eigenvectors were fixed your components will have an ambiguity (each of them individually) of a complex factor: just as well say that you don't know them (except if they happen to be 0).

Happily, you can hold the same reasoning for B :-) so I would say that a possible answer to the question will always be, that in a suitably scaled set of eigenvectors of B, psi will always take on the components {1,1,1,1,...1}


cheers,
Patrick.

 

[broegger]

Sorry, I'm wasting your time. The situation in 1) is this:

An electron moving in a Coulomb-field from a proton, is in the following state (at time t=0)

$$|\phi,t=0> = \tfrac4{5}|100> + \tfrac{3i}5|211>$$​

where $$|nlm>$$ is the usual energy eigenstates of the hydrogenatom. They are also eigenstates of angularmomentum:

$$L^2|nlm>=l(l+1)\hslash^2|nlm>$$

$$L_z|nlm>=m\hslash|nlm>$$
​

The questions are now:

a) Calculate the expectation value <E> in the state $$|\phi,t=0>$$.

b) Calculate the expectation values for L^2 and L_z in the state $$|\phi,t=0>$$.

c) Calculate the expectation value for L_x in the state $$|\phi,t=0>$$.

It's c) I'm having trouble with. Hope this helps.

 

[dextercioby]

$$\langle \hat{L}_{x}\rangle_{|\psi\rangle} =:\langle\psi|\hat{L}_{x}|\psi\rangle$$

Okay?

Now,use the fact that

$$\hat{L}_{x}=\frac{1}{2}\left(\hat{L}_{+}+\hat{L}_{-}\right)$$

and the action of $\hat{\mbox{L}}_{+}$ and $\hat{\mbox{L}}_{-}$ on an arbitrary hydrogenoid wavefunction $|n,l,m\rangle$ (see textbook) and the orthonormalization of the states

$$\langle n,l,m|n',l',m'\rangle =\delta_{nn'}\delta_{ll'}\delta_{mm'}$$

Daniel.

 

[broegger]

Yep, I with you now :) Thank you both.

 

[shyboy]

I assume that eigenvectors of both A and B provide the complete basis. (?)
well, about matrices

you have
|c>=|A><A|c>

you need to find

|c>=|B><B|c>

where |A> and |B> - eigenvectors

then

<b|c>=<B|A><A|c>;

 

[dextercioby]

I don't follow.What and who are those A,B,a,b,,blah,blah,blah...?

Daniel.

 

[shyboy]

that is an old linear algebra in QM notation. I assume that eigenvectors of each matrix forms an ortogonal basis.

You have vector in the basis of the A matrix

$$\vec\psi=a_1 \vec{a_1}+a_2 \vec{a_2}+...$$ (1)

you need to find

$$\vec\psi=b_1 \vec{ b_1}+b_2 \vec{ b_2}+...$$ (2)

so multiply (1) by $$\vec{b_1}, \vec{b_2}...$$ and so on.

$$b_1 =a_1 (\vec {a_1}\vec{ b_1})+a_2 (\vec {a_1}\vec{ b_1})+...$$

 

[dextercioby]

Okay.What relevance does it have here?

Daniel.

 

[shyboy]

well,

Say, I have a state and two operators A and B represented as matrices. Now, $$\psi$$ is given as a linear combination of the eigenstates of A

that was (1)

and I want to express them as a linear combination of the eigenstates of the operator B instead.

that was (2)

 

[dextercioby]

Are u hijacking the thread?This was the context

Two quick ones :)

--------------------------------------------------------------------------------

Hi, two questions:

1) How can I find the expectation value of the x-component of the angular momentum, , when I know and ?

2) Say, I have a state and two operators A and B represented as matrices. Now, is given as a linear combination of the eigenstates of A and I want to express them as a linear combination of the eigenstates of the operator B instead. How do I do that?

Thanks :)


[end quote]


Daniel.